Nonsmooth contact mechanics. Spring school. Aussois 2010

Transcription

1 1/43 Nonsmooth contact mechanics. Spring school. Aussois 2010

2 Formulation of nonsmooth dynamical systems Measure differential inclusions Basics on Mathematical properties Formulation of unilateral contact, Coulomb s friction and impacts. Flavor of enhanced nonsmooth laws 2/43

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4 4/43 Definition (Lagrange s equations) where d dt L(q, v) v i q(t) R n generalized coordinates, «L(q, v) = Q i (q, t), i {1... n}, (1) q i v(t) = dq(t) R n generalized velocities, dt Q(q, t) R n generalized forces L(q, v) IR Lagrangian of the system, L(q, v) = T(q, v) V(q), together with T(q,v) = 1 2 vt M(q)v, kinetic energy, M(q) R n n mass matrix, V(q) potential energy of the system,

5 Lagrange equations M(q) dv + N(q, v) = Q(q, t) qv(q) (2) dt where 2 3 N(q, v) = 4 1 X M ik + M il M kl, i = 1... n5 the nonlinear 2 q k,l l q k q i inertial terms i.e., the gyroscopic accelerations Internal and external forces which do not derive from a potential where M(q) dv dt + N(q, v) + F int(t, q, v) = F ext(t), (3) F int : R n R n R R n non linear interactions between bodies, F ext : R R n external applied loads. Linear time invariant (LTI) case M(q) = M IR n n mass matrix F int (t,q, v) = Cv + Kq, C IR n n is the viscosity matrix, K IR n n is the stiffness matrix. 5/43

6 6/43 Definition (Smooth multibody ) 8 < M(q) dv + F(t, q, v) = 0, dt : v = q where F(t,q, v) = N(q, v) + F int (t, q, v) F ext(t) Definition (Boundary conditions) Initial Value Problem (IVP): (4) t 0 R, q(t 0 ) = q 0 R n, v(t 0 ) = v 0 R n, (5) Boundary Value Problem (BVP): (t 0, T) R R, Γ(q(t 0 ), v(t 0 ), q(t),v(t)) = 0 (6)

7 7/43 Perfect bilateral constraints, joints, liaisons and spatial boundary conditions Bilateral constraints Finite set of m bilateral constraints on the generalized coordinates : h(q, t) = ˆh j (q, t) = 0, j {1... m} T. (7) where h j are sufficiently smooth with regular gradients, q(h j ). Configuration manifold, M(t) M(t) = {q(t) R n, h(q, t) = 0}, (8) Tangent and normal space Tangent space to the manifold M at q T M (q) = {ξ, h(q) T ξ = 0} (9) Normal space as the orthogonal to the tangent space N M (q) = {η, η T ξ = 0, ξ T M } (10)

8 8/43 Bilateral constraints as inclusion Definition (Perfect bilateral holonomic constraints on the smooth ) 8 q = v >< M(q) dv + F(t, q, v) = r dt >: r N M (q) where r is the generalized force or generalized reaction due to the constraints. (11) Remark The formulation as an inclusion is very useful in practice The constraints are said to be perfect due to the normality condition. When M = {q(t) R n, h(q, t) = 0}, the multipliers µ R m can be intoduced and we get r = qh(q, t) µ

9 9/43 Perfect unilateral constraints Unilateral constraints Finite set of ν unilateral constraints on the generalized coordinates : g(q, t) = [g α(q, t) 0, α {1... ν}] T. (12) Admissible set C(t) C(t) = {q R n, g α(q, t) 0, α {1... ν}}. (13) Normal cone to C(t) N C(t) (q(t)) = ( y R n, y = X α λ α g α(q, t), λ α 0, λ αg α(q, t) = 0 ) (14)

10 10/43 Unilateral constraints as an inclusion Definition (Perfect unilateral constraints on the smooth ) 8 q = v >< M(q) dv + F(t, q, v) = r dt >: r N C(t) (q(t)) where r it the generalized force or generalized reaction due to the constraints. (15) Remark The unilateral constraints are said to be perfect due to the normality condition. Notion of normal cones can be extended to more general sets. see (Clarke, 1975, 1983 ; Mordukhovich, 1994) When C(t) = {q R n, g α(q, t) 0, α {1... ν}}, the multipliers λ R m such that r = T q g(q, t) λ.

11 11/43 Smooth as a DI Differential Inclusion» M(q) dv + F(t,q, v) N C(t) (q(t)), (16) dt with Remark q = v. The right hand side is neither bounded (and then nor compact). The inclusion and the constraints concern the second order time derivative of q. Standard Analysis of DI does no longer apply.

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13 13/43 Nonsmooth Lagrangian Fundamental assumptions. The velocity v = q is of Bounded Variations (B.V) The equation are written in terms of a right continuous B.V. (R.C.B.V.) function, v + such that q is related to this velocity by v + = q + (17) Z t q(t) = q(t 0 ) + v + (t)dt t 0 (18) The acceleration, ( q in the usual sense) is hence a differential measure dv associated with v such that Z dv(]a,b]) = dv = v + (b) v + (a) (19) ]a,b]

14 14/43 Nonsmooth Lagrangian Definition (Nonsmooth Lagrangian ) 8 >< M(q)dv + F(t, q, v + )dt = di (20) >: v + = q + where di is the reaction measure and dt is the Lebesgue measure. Remarks The nonsmooth contains the impact equations and the smooth evolution in a single equation. The formulation allows one to take into account very complex behaviors, especially, finite accumulation (Zeno-state). This formulation is sound from a mathematical Analysis point of view. (Schatzman, 1973, 1978 ; Moreau, 1983, 1988)

15 15/43 Nonsmooth Lagrangian Decomposition of measure where j dv = γ dt+ (v + v ) dν+ dv s di = f dt+ p dν+ di s (21) γ = q is the acceleration defined in the usual sense. f is the Lebesgue measurable force, v + v is the difference between the right continuous and the left continuous functions associated with the B.V. function v = q, dν is a purely atomic measure concentrated at the time t i of discontinuities of v, i.e. where (v + v ) 0,i.e. dν = P i δt i p is the purely atomic impact percussions such that pdν = P i p iδ ti dv S and di S are singular measures with the respect to dt + dη.

16 16/43 Impact equations and Smooth Lagrangian Substituting the decomposition of measures into the nonsmooth Lagrangian, one obtains Definition (Impact equations) or M(q)(v + v )dν = pdν, (22) M(q(t i ))(v + (t i ) v (t i )) = p i, (23) Definition (Smooth between impacts) or M(q)γdt + F(t, q, v)dt = fdt (24) M(q)γ + + F(t, q, v + ) = f + [dt a.e.] (25)

17 of second order Definition (Moreau (1983, 1988)) A key stone of this formulation is the inclusion in terms of velocity. Indeed, the inclusion (15) is replaced by the following inclusion 8 M(q)dv + F(t, q, v + )dt = di >< v + = q + >: di N TC (q)(v + ) (26) Comments This formulation provides a common framework for the nonsmooth containing inelastic impacts without decomposition. Foundation for the time stepping approaches. 17/43

18 18/43 of second order Comments The inclusion concerns measures. Therefore, it is necessary to define what is the inclusion of a measure into a cone. The inclusion in terms of velocity v + rather than of the coordinates q. Interpretation Inclusion of measure, di K Case di = r dt = fdt. Case di = p i δ i. Inclusion in terms of the velocity. Viability Lemma If q(t 0 ) C(t 0 ), then f K (27) p i K (28) v + T C (q), t t 0 q(t) C(t),t t 0 The unilateral constraints on q are satisfied. The equivalence needs at least an impact inelastic rule.

19 19/43 of second order The Newton-Moreau impact rule where e is a coefficient of restitution. di N TC (q(t))(v + (t) + ev (t)) (29) Velocity level formulation. Index reduction 0 y λ 0 λ N R +(y) λ N TR +(y)(ẏ) if y 0 then 0 ẏ λ 0 (30)

20 of second order The case of C is finitely represented C = {q M(t),g α(q) 0, α {1... ν}}. (31) Decomposition of di and v + onto the tangent and the normal cone. di = X α T q gα(q)dλα (32) U + α = qg α(q)v +, α {1... ν} (33) Complementarity formulation (under constraints qualification condition) dλ α N TIR+ (g α)(u + α) if g α(q) 0, then 0 U + α dλ α 0 (34) The case of C is IR + is replaced by di N C (q) 0 q di 0 (35) di N TC (q)(v + ) if q 0, then 0 v + di 0 (36) 20/43

21 21/43 of second order Summary for perfect schleronomic constraints 8 M(q)dv + F(t,q, v + )dt = di v + = q + >< di = H(q)dλ (37) U + = H(q) T v + >: if g α(q) 0, then 0 U α + dλ α 0 where H(q) is the transpose of the Jacobian matrix of the constraints, H(q) = qg(q).

22 22/43 in Newton Euler Formalism Classical Newton-Euler formalism The velocity of a rigid body is represented with v G R 3 the velocity of the center of mass expressed in a Galilean reference frame R 0, Ω R 3 the angular velocity expressed in a frame attached to the solid R, called the body frame. Rotation matrix and angular velocity vector By defining the rotation matrix R SO + (3) from R 0 to R, the angular velocity is given by Ω = R T Ṙ or equivalently Ṙ = R Ω. (38) where the matrix Ω is defined by Ωx = Ω x, for all x R 3.

23 23/43 in Newton Euler Formalism Smooth Newton-Euler Equations From the Fundamental Principle of, the Newton Euler equations are obtained as where 8 >< >: M v G = F ext(x G, v G, Ω, R), I Ω + Ω IΩ = M ext(x G, v G, Ω, R), ẋ G = v G, Ṙ = R Ω, R 1 = R T, det(r) = 1. x G is the position of the center of mass, M = mi 3 3 is the mass matrix and I the constant inertia matrix, (39) F ext is the vector of external forces expressed in R 0 M ext is the vector of external moments expressed in R Another angular velocity vector The Newton Euler equations can be also expressed in terms of the angular velocity ω = RΩ that is the expression of the angular velocity in R 0.

24 24/43 in Newton Euler Formalism General Formulation Choosing q = [x G, R] T and v = [v G, Ω], the Newton Euler equations fits within the general framework 8 q = T(t,q)v, (40a) >< M(q) v + F(t,q, v) = T T (t,q)r = T T (t,q)h(q)µ (40b) >: h(q) = 0 (40c) where H(q) = T q h(q) T(t,q) is the operator that links the velocity to the time derivative of the parameters, h(q) = 0 are the constraints for the configuration manifold R SO + (3)

25 25/43 in Newton Euler Formalism Parametrization of rotations The choice q = [x G, R] T R 12 is not well suited for numerical computation. Generally, the rotation matrix is parametrized by a set of parameters, Θ such that R = R(Θ) and we get Examples of parametrization: ω = P(Θ) Θ or Ω = Q(Θ) Θ. geometrical description angles : Euler angles, Cardan/Bryant Angles, Rodrigues parameters, direct cosines, unitary quaternions, Cartesian oration vector Conformal rotation vector, linear parameters,...

26 26/43 in Newton Euler Formalism Smooth DI for Newton Euler Formalism 8» >< M(q) dv + F(t, q, v) T T (q, t)n C(t) (q(t)) dt >: q = T(q,t)v The case of C is finitely represented we get (41) C = {q R n, g α(q) 0, α I, g α(q) = 0, α E}. (42) 8 q =» T(q,t)v M(q) dv + F(t,q, v) T T (q, t)r dt >< r = H(q)λ U = H T (q)t(q,t)v (43) g α(q) = 0, α E >: 0 g α(q) λ α 0, α E

27 27/43 in Newton Euler Formalism Measure DI for Newton Euler Formalism 8 [M(q)dv + F(t, q, v)dt] = T T (q, t)di >< di N TC (q)(v + ) >: q + = T(q,t)v + (44)

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29 29/43 Academic examples The bouncing Ball and the linear impacting oscillator m f q q 0 (a) Bouncing ball example 0 m (b) Linear Oscillator example Figure: Academic test examples with analytical solutions

30 NonSmooth Multibody Systems (NSMBS) 1 Exact Solution. Bouncing Ball Example position velocity time (s) Figure: Analytical solution. Bouncing ball example 30/43

31 NonSmooth Multibody Systems (NSMBS) 4 3 Exact Solution. Linear Oscillator Example position velocity time (s) Figure: Analytical solution. Linear Oscillator 30/43

32 31/43 of second order Example (The Bouncing Ball) f(t) z R θ g O h x ) Figure: Two-dimensional bouncing ball on a rigid plane

33 31/43 of second order Example (The Bouncing Ball) In our special case, the model is completely linear: 2 3 q = M(q) = N(q, q) = 4 z x 5 (45) θ 2 4 m m 0 5 where I = I 5 mr2 (46) (47) F int (q, q, t) = F ext(t) = (48) mg f (t) (49) 0 0

34 31/43 of second order Example (The Bouncing Ball) Kinematics Relations The unilateral constraint requires that : C = {q, g(q) = z R h 0} (45) so we identify the terms of the equation the equation (31) di = [1, 0, 0] T dλ 1, (46) 2 3 ż U + 1 = [1, 0, 0] 4 ẋ 5 = ż (47) θ Nonsmooth laws The following contact laws can be written, 8 >< if g(q) 0, then 0 U + + eu dλ 1 0 >: if g(q) 0, dλ 1 = 0 (48)

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36 33/43 Local coordinates system at contact Lagrangian approach of constraints is not sufficient. The elegant Lagrangian approach of unilateral constraints and their associated multipliers is not sufficient for describing more complex behavior of the contact : The Lagrange multipliers have no physical dimensions The constraints can be multiplied by a positive constant. For a mechanical description of the behaviour of the contact interface, a (set-valued) force laws needs to be introduced together with a coordinate systems at contact.

37 34/43 Local coordinates system at contact Body O n C s t Body O P Definition of a contact frame Assume that we have defined P and P proximal points between O and O n an outward unit normal vector along P P t and s two unit tangent vectors g(q) a gap function, i.e., the signed distance P P n t s P Body O Remark This definition is not trivial for a nonsmooth or nonconvex surfaces.

38 35/43 Local coordinates system at contact Relative local velocity The relative local velocity U is defined by and is decomposed in the frame (P,n,t,s) as U = V P V P (49) U = U Nn + U T, U N R, U T R 2 (50) Link with the gap function The derivative with respect to time of the gap function t g(q(t)) is the normal relative velocity U N ġ( ) = U N( ) = g T (q)v( ) (51) Local reaction force at contact The relative local velocity R acts from O to O and is also decomposed as U = R Nn + R T, R N R,R T R 2 (52)

39 Local coordinates system at contact Relations with global/generalized coordinates Is is assumed that there exists a relation between the local relative velocity U and the velocity of bodies v such that By duality (expressed in terms of power) we get U = H T (q)v (53) r = H(q)R (54) Unilateral contact in terms of local variables if g(q) 0, then 0 U N R N 0 (55) 36/43

40 37/43 Coulomb s friction C U T R N R n t P s R T Figure: Coulomb s friction. The sliding case. Definition (Coulomb s friction) Coulomb s friction says the following. If g(q) = 0 then: 8 If U T = 0 then R C >< If U T 0 then R T(t) = µ R N and there exists a scalar a 0 >: such that R T = au T (56) where C = {R, R T µ R N } is the Coulomb friction cone

41 38/43 Coulomb s friction Definition (Coulomb s friction as an inclusion into a disk) Let us introduce the following inclusion (Moreau, 1988), using the indicator function ψ D ( ): U T ψ D (R T) (57) where D = {R T, R T(t) µ R N } is the Coulomb friction disk Definition (Coulomb s friction as a variational inequality (VI)) Then (56) appears to be equivalent to 8 < R T D : U T, z R T 0 for all z D (58) and to R T = proj D [R T ρu T], for all ρ > 0 (59)

42 Definition (Coulomb s Friction as a Second Order Cone Complementarity Problem) Let us introduce the modified velocity b U defined by bu = [U N + µ U T, U T] T. (60) This notation provides us with a synthetic form of the Coulomb friction as b U ψ C (R), (61) or C b U R C. (62) where C = {v IR n r T v 0, r C} is the dual cone. 39/43

43 40/43 Coulomb s friction C R N U T U b µ U T n R t P s µ U T R T U T b U C Figure: Coulomb s friction and the modified velocity b U. The sliding case.

44 41/43 Coulomb s friction with impacts It is for instance proposed in (Moreau, 1988) to extend (56) (??) to densities, i.e. to impulses with a tangential restitution 8 P >< N ψ 1 IR ( 1 + ρ U+ (t) + ρ N 1 + ρ U (t)) N >: P T ψ D ( τ U+ T (t) + τ 1 + τ U T (t)). with ρ and τ are constants with values in the interval [0, 1] or 8 < : P N ψ IR (U + N (t) + enu N (t)) (63) P T ψ D (U+ T (t) + etu T (t)) (64) where e N [0, 1) and e T ( 1, 1).

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46 43/43 Thank you for your attention.

47 43/43 F.H. Clarke. Generalized gradients and its applications. Transactions of A.M.S., 205: , F.H. Clarke. Optimization and Nonsmooth analysis. Wiley, New York, B.S. Mordukhovich. Generalized differential calculus for nonsmooth ans set-valued analysis. Journal of Mathematical analysis and applications, 183: , J.J. Moreau. Approximation en graphe d une évolution discontinue. RAIRO Analyse numérique/ Numerical Analysis, 12:75 84, J.J. Moreau. Liaisons unilatérales sans frottement et chocs inélastiques. Comptes Rendus de l Académie des Sciences, 296 serie II: , J.J. Moreau. Unilateral contact and dry friction in finite freedom. In J.J. Moreau and Panagiotopoulos P.D., editors, Nonsmooth mechanics and applications, number 302 in CISM, Courses and lectures, pages CISM 302, Spinger Verlag, Formulation mathematiques tire du livre Contacts mechanics. M. Schatzman. Sur une classe de problmes hyperboliques non linaires. Comptes Rendus de l Académie des Sciences Srie A, 277: , M. Schatzman. A class of nonlinear differential equations of second order in time. Nonlinear Analysis, T.M.A, 2(3): , 1978.